Using the first three terms of the Taylor series expansion of f (x) = x 3 f(x) = \sqrt[3]{x} f (x) = 3 x centered at x = 8 x = 8 x = 8, approximate 8.1 3: \sqrt[3]{8.1}: 3 8. 1 : f (x) = x 3 ≈ 2 + (x − 8) 12

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Let's begin by looking at its definition and an example: Let f be a function that has derivatives of all orders at x = a. The Taylor Series for f at x = a is. T f ( x) = f ( a) + f ′ ( a) ( x − a) + f ″ ( a) 2! ( x

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Example: Assume a finite Taylor series approximation that converges everywhere for a given function !(#)and you are given the following information: !1=2;!)(1)=−3;!))(1)=4;!-1=0∀0≥3

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